Solution;
Given ;
T(x1,x2)=[x1−x20]
A map T is linear if;
T(x)=Ax
A is the matrix of transformation;
Here;
A=[10−10]
Clearly;
T(x1,x2)=[10−10] [x1x2] =(x1−x2,0)
Therefore the map is a linear transformation.
And;
B=[(−1,1),(0,1)]
Hence;
TB(−1,1)=[−1−10]=[−20]
TB(0,1)=[0−10]=[−10]
Hence;
TB=T[(−1,1),(0,1)]=[(−2,0),(−1,0)]
Let;
(−2,0)=a1(−1,1)+b1(0,1)
(−2,0)=(−a1,a1)+(0,b1)
(−2,0)=(−a1,a1+b1)
From which;
−2=−a1 ∴a1=2
Also;
O=a1+b1 ;
b1=−a1=−2
Now,let;
(−1,0)=a2(−1,1)+b2(0,1)
(−1,0)=(−a2,a2)+(0,b2)
(−1,0)=(−a2,a2+b2)
From which;
−1=−a2,∴a2=1
Also;
0=a2+b2 ;
b2=−a2=−1
The coefficient matrix A,is ;
A=[21−2−1]
Hence,the matrix of transformation of T relative to B is AT;
AT=[2−21−1]
Comments
Leave a comment